%%
%% hw3.tex
%% 
%% Made by alex
%% Login   <alex@tomato>
%% 
%% Started on  Mon Oct  3 12:13:03 2011 alex
%% Last update Mon Oct  3 12:13:03 2011 alex
%%
\section*{Homework 3}\renewcommand{\leftmark}{Homework 3}\phantomsection\addcontentsline{toc}{section}{Homework 3}
\begin{exercise}
Let $f$, $g$ be two analytic functions in a domain ${\cal U}$
bounded by a closed curve $\gamma$, continuous in the closure of
${\cal U}$, and let $0 < \|g(z)\| < \|f (z)\|$ for all
$z\in\gamma$. Prove that the numbers(with multiplicities) of
solutions of equations $f (z) = g(z)$ and $f (z) = 0$ are the
same. 
\end{exercise}
\begin{exercise}
How many zeroes does $z^{6} - 4z^{5} + z^{2} - 1$ have in the disk $D = \{z \lst \|z\| < 1\}$?
\end{exercise}
\begin{exercise}
Show that if $p(z) = z^{n} + a_{n-1} z^{n-1} + \dots + a_{1} z + a_{0}$ then there must be a point $z$
with $\|z\| = 1$ and $p(z)\geq1$.
\end{exercise}
\begin{exercise}
Let $\displaystyle g_{n}=\sum^{n}_{k=0}\frac{z^{k}}{k}$. Let
$D(0, R)$ be the disk of radius $R > 0$. Show that for $n$ large
enough $g_{n}$ has no zeroes in $D(0; R)$. 
\end{exercise}
\begin{exercise}
Let $\alpha$ be a complex number. 

(a) How many values at a generic $z$ does the function
$z (= \E^{\alpha \log z} )$ have? (The answer depends on $\alpha$.) 

(b) Describe the Riemann surface of this function. 

(c) In terms of $\|z\|$ and $\arg z$ (and standard elementary
functions from Real Analysis) list all the values of $z^{\I}$. 
\end{exercise}
